Question: Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle CBD = 7x + 14$, and $ m \angle ABC = 3x - 4$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Solution: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {3x - 4} + {7x + 14} = {90}$ Combine like terms: $ 10x + 10 = 90$ Subtract $10$ from both sides: $ 10x = 80$ Divide both sides by $10$ to find $x$ $ x = 8$ Substitute $8$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 7({8}) + 14$ Simplify: $ {m\angle CBD = 56 + 14}$ So ${m\angle CBD = 70}$.